A novel one-dimensional refined model for the analysis of laminated structures is presented in this paper. The Layer Wise approach in conjunction with the Carrera Unified Formulation (CUF) enables to create hierarchical refined models which are capable of detecting both local and global mechanical behaviours. In the framework of the CUF, the FEM matrices are obtained in the form of fundamental nuclei, which are independent of the theory adopted for the model and the FEM discretization along the beam axis. A hierarchical set of expansions above the cross-section is developed by adopting Legendre-class polynomials for the definition of the beam theory kinematics. In this way, the polynomial order of the expansion can be introduced as a free-parameter, enabling to vary the accuracy of the theory of structure in a straightforward manner. Compact cross-ply laminates with different number of layers, as well as more complex composite cases like box beams have been adopted to test the Hierarchical Legendre Expansions (HLE) adapted to the Layer Wise approach. The model has been verified through published literature and also through the use of the commercial software MSC Nastran.

This paper discusses classical and refined beam and plate theories based on the Carrera Unified Formulation (CUF). Attention is focussed on (but not limited to) a new refined beam element with enhanced kinematics based on Legendre polynomial expansions of the primary mechanical variables. By employing CUF, the governing equations and the related finite element arrays are written in a hierarchical, compact and general manner. Readily, these characteristics are used to arbitrarily tune the finite element model at the cross-sectional level, by locally enriching the theory kinematics up to the desired accuracy. The uncompromising accuracy of the present beam model is demonstrated by considering various numerical examples, including solid and thin-walled beams with open and close cross-sections as well as plate structures. The results are compared with those from classical and already established refined CUF models. Eventually, three-dimensional elasticity solutions by the commercial tool MSC Nastran are also given to underline the high accuracy of the present methodology. The numerical efficiency and the capabilities of the Legendre-based CUF beam models to deal with complex structures with no geometrical approximations result clear from the analyses conducted.

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A class of mixed interpolated beam elements is introduced in this paper under the framework of the Carrera Unified Formulation to eliminate the detrimental effects due to shear locking. The Mixed Interpolation of Tensorial Components (MITC) method is adopted to generate locking-free displacement-based beam models using general 1D finite elements. An assumed distribution of the transverse shear strains is used for the derivation of the virtual work, and the full Gauss-Legendre quadrature is used for the numerical computation of all the components of the stiffness matrix. Linear, quadratic, and cubic beam elements are developed using the unified formulation and applied to linear static problems including compact, laminated, and thin-walled structures. A comprehensive study of how shear locking affects general beam elements when different classical integration schemes are used is presented, evidencing the outstanding capabilities of the MITC method to overcome this numerical issue. Refined beam theories based on the expansion of pure and generalized displacement variables are implemented making use of Lagrange and Legendre polynomials over the cross-sectional domain, allowing one to capture complex states of stress with a 3D-like accuracy. The numerical examples are compared to analytic, numerical solutions from the literature, and commercial software solutions, whenever it is possible. The efficiency and robustness of the proposed method demonstrated throughout all the assessments, illustrating that MITC elements are the natural choice to avoid shear locking and showing an unprecedent accuracy in the computation of transverse shear stresses for beam formulations.

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